Coarse homotopy groups

In this note on coarse geometry we revisit coarse homotopy. We prove that coarse homotopy indeed is an equivalence relation, and this in the most general context of abstract coarse structures. We introduce (in a geometric way) coarse homotopy groups. The main result is that the coarse homotopy groups of a cone over a compact simplicial complex coincide with the usual homotopy groups of the underlying compact simplicial complex. To prove this we develop geometric triangulation techniques for cones which we expect to be of relevance also in different contexts.     

A Braided Simplicial Group

By studying the braid group action on Milnor's constructionof the 1-sphere, we show that the general higher homotopy groupof the 3-sphere is the fixed set of the pure braid group actionon certain combinatorially described groups. This establishesa relation between the braid groups and the homotopy groupsof the sphere. 2000 Mathematical Subject Classification: 20F36, 55P35, 55Q05,55Q40, 55U10.     

A Banach Space Whose Operator Algebra has K0-Group Q

We construct a Banach space X whose algebra of bounded linearoperators has K0-group the additive group of rational numbers.We achieve this using a Gowers-Maurey space. 2000 Mathematical Subject Classification: 20F36, 55P35, 55Q05,55Q40, 55U10     

A nontrivial product in the stable homotopy groups of spheres

Let A be the mod p Steenrod algebra for p an arbitrary odd prime. In 1962, Li-uleviciusdescribed hi and bk in Ext (A|*,*) (Zp, Zp) having bigrading (1,2pi(p-1))and (2,2pk+1 x(p - 1)), respectively. In this paper we prove that for p ≥ 7,n ≥ 4 and 3 ≤ s < p - 1, (Zp,Zp) survives to E∞ in the Adams spectral sequence, where q = 2(p - 1).     

Bar complex, configuration spaces and finite type invariants for braids

We show that the bar complex of the configuration space of ordered distinct points in the complex plane is acyclic. The 0-dimensional cohomology of this bar complex is identified with the space of finite type invariants for braids. We construct a universal holonomy homomorphism from the braid group to the space of horizontal chord diagrams over Q, which provides finite type invariants for braids with values in Q.     

R~n中构形和单纯复形的一个最优度量不等式

对于R~n中一般位置的点构形,定义了第r个极小凸包距离的概念,证明了极小凸包距离和极小点-超平面距离之间的一个最优不等式.该不等式的一个直接推论是:对于R~n中一个k-维单纯复形K,我们能用其顶点集的极小点-超平面距离下估计K的Gromov-Guth厚度.进一步,在每一个维数k,构造了例子说明该下界几乎是最优的.     

Spaces of Maps into Classifying Spaces for Equivariant Crossed Complexes, II: The General Topological Group Case

The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using detailed results on the appropriate Eilenberg–Zilber theory, and of its relation to simplicial homotopy coherence. Again, our results give information not just on the homotopy classification of certain equivariant maps, but also on the weak equivariant homotopy type of the corresponding equivariant function spaces.     

Moment-angle复形W_(k,d)轨道构型空间的欧拉示性数

给定方体的典范单纯剖分.将单纯复形K的重心充分K′逐片线性嵌入高维方体中,从而得到K对应的方体复形cub(K).由cub(K)的构造,计算了cub(K)的f-向量.cub(K)上可以定义moment-angle复形W_(K,d).将W_(K,d)放入轨道构型空间的框架中,得到轨道构型空间FG(W_(K,d),n).利用著名的Inclusion-exclsion原理和cub(K)的f-向量,计算出了轨道构型空间FG(W_(K,d),n)的欧拉示性数,并且给出了一种计算W_(K,d)欧拉示性数的新方法.     

On the homotopy fixed point problem for free loop spaces and other function complexes

Let G be a finite group, let X and Y be finite G-complexes, and suppose that for each K G, Y ^K is dim(X ^K)-connected and simple. G acts on the function complex F(X, Y) by conjugation of maps. We give a complete analysis of the homotopy fixed point set of the space F(X, Y). As a corollary, we are able to analyze at a prime p, the homotopy fixed point set of the circle action on X, where X denotes the free loop space of X, and X is a simply connected finite complex.Supported in part by NSF DMS 86-02430.To A. Grothendieck on the occasion of his sixtieth birthday     

Two New Families in the Stable Homotopy Groups of Sphere and Moore Spectrum

This paper proves the existence of an order p element in the stable homotopy group of sphere spectrum of degree pnq pmq q - 4 and a nontrivial element in the stable homotopy group of Moore spectum of degree pnq pmq q - 3 which are represented by h0(hmbn-1-hnbm-1) and i*(h0hnhm) in the E2-terms of the Adams spectral sequence respectively, where p≥7 is a prime, n≥m 2≥4, q = 2(p - 1).     

On homotopy groups of quandle spaces and the quandle homotopy invariant of links

For a quandle X, the quandle space BX is defined, modifying the rack space of Fenn, Rourke and Sanderson (1995) [13], and the quandle homotopy invariant of links is defined in Z[π₂(BX)], modifying the rack homotopy invariant of Fenn, Rourke and Sanderson (1995) [13]. It is known that the cocycle invariants introduced in Carter et al. (2005) [3], Carter et al. (2003) [5], Carter et al. (2001) [6] can be derived from the quandle homotopy invariant.In this paper, we show that, for a finite quandle X, π₂(BX) is finitely generated, and that, for a connected finite quandle X, π₂(BX) is finite. It follows that the space spanned by cocycle invariants for a finite quandle is finitely generated. Further, we calculate π₂(BX) for some concrete quandles. From the calculation, all cocycle invariants for those quandles are concretely presented. Moreover, we show formulas of the quandle homotopy invariant for connected sum of knots and for the mirror image of links.     

A Four-filtrated may spectral sequence and its applications

In this paper, we introduce a four-filtrated version of the May spectral sequence (MSS), from which we study the general properties of the spectral sequence and give a collapse theorem. We also give an efficient method to detect generators of May E ₁-term E ₁ ^s,t,b,* for a given (s, t, b, *). As an application, we give a method to prove the non-triviality of some compositions of the known homotopy elements in the classical Adams spectral sequence (ASS). This research is partially supported by the National Natural Science Foundation of China (Nos. 10501045, 10771105) and the Fund of the Personnel Division of Nankai University     

2-Cohomologies of the groups SL(n, q)

Let G = SL(n, q), where q is odd, V be a natural module over G, and L = S²(V) be its symmetric square. We construct a 2-cohomology group H²(G, L). The group is one-dimensional over F _q if n = 2 and q ≠ 3, and also if (n, q) = (4, 3). In all other cases H²(G, L) = 0. Previously, such groups H²(G, L) were known for the cases where n = 2 or q = p is prime. We state that H²(G, L) are trivial for n ⩾ 3 and q = p^m, m ⩾ 2. In proofs, use is made of rather elementary (noncohomological) methods. __________ Translated from Algebra i Logika, Vol. 47, No. 6, pp. 687–704, November–December, 2008.     

Compact open topology and CW homotopy type

The class of spaces having the homotopy type of a CW complex is not closed under formation of function spaces. In 1959, Milnor proved the fundamental theorem that, given a space and a compact Hausdorff space X, the space Y^X of continuous functions X→Y, endowed with the compact open topology, belongs to . P.J. Kahn extended this in 1982, showing that if X has finite n-skeleton and π_k(Y)=0, k>n.

Using a different approach, we obtain a further generalization and give interesting examples of function spaces where is not homotopy equivalent to a finite complex, and has infinitely many nontrivial homotopy groups. We also obtain information about some topological properties that are intimately related to CW homotopy type.

As an application we solve a related problem concerning towers of fibrations between spaces of CW homotopy type.     

Combinatorial Harmonic Maps and Discrete-group Actions on Hadamard Spaces

In this paper we use the combinatorial harmonic map theory to study the isometric actions of discrete groups on Hadamard spaces. Given a finitely generated group acting by automorphisms, properly discontinuously and cofinitely on a simplicial complex and its isometric action on a Hadamard spaces, we formulate criterions for the action to have a global fixed point.Dedicated to Professor Takushiro Ochiai on his 60th birthday.